The determined property of Baire in reverse math
Abstract
We define the notion of a determined Borel code in reverse math, and consider the principle $DPB$, which states that every determined Borel set has the property of Baire. We show that this principle is strictly weaker than $ATR$. Any $\omega$model of $DPB$ must be closed under hyperarithmetic reduction, but $DPB$ is not a theory of hyperarithmetic analysis. We show that whenever $M\subseteq 2^\omega$ is the secondorder part of an $\omega$model of $DPB$, then for every $Z \in M$, there is a $G \in M$ such that $G$ is $\Delta^1_1$generic relative to $Z$.
 Publication:

arXiv eprints
 Pub Date:
 September 2018
 arXiv:
 arXiv:1809.03940
 Bibcode:
 2018arXiv180903940A
 Keywords:

 Mathematics  Logic;
 03B30
 EPrint:
 Greatly expanded introduction as requested by referee